LOUIS  CLARK  VANUXEM  FOUNDATION 

THE  THEORY  OF 
PERMUTABLE  FUNCTIONS 


BY 

VITO  VOLTERRA 

Professor  of  Mathematical  Physics  in  the 
University  of  Rome 


Lectures  Delivered  at  Princeton  University 
October,  1912 


PRINCETON    UNIVERSITY   PRESS 

PRINCETON 

LONDON  :  HUMPHREY  MILFORD 

OXFORD  UNIVERSITY  PRESS 

I9I5 


Published  April,  191 5 


LECTURE  I 


302503 


LECTURE  I 

1.  We  shall  begin  with  quite  elementary  and 
general  notions. 

First,  let  us  recall  the  properties  of  a  sum 

n 

«1  +  ^^2  +      •     •     •      +   ((n  =  ^i  ^i  • 

I 

This  operation  is  both  associative  and  com- 
mutative, that  is, 

[a  +  b)  +  c  =  a+(b-i-c) 
and 

a  -\r  h  =^  h  +  a  . 

Now  we  can  jDass  from  a  sum  to  an  integral 
by  a  well-known  limiting  process.  For  the 
sake  of  simplicity,  we  shall  make  use  of  the 
definition  of  Riemann:  Given  a  function  f  (<.v) 
which  is  defined  over  an  interval  ah,  we  sub- 
divide the  interval  ah  into  n  parts  //i,  h-,  hs, 
h^  hn .  Corresponding  to  every  in- 
terval hi  we  then  take  some  value  /,  of  /(<r) 
lying  between  the  upper  and  lower  limits  of 
/(.r)  on  hj ,  and  we  form  the  sum 

i  /;  ^. . 
1 


4  THEORY  OF  PERMUTABLE  FUNCTIONS 

Now  suppose  we  allow  hi,  /lo,  Jh, h ,(  to 

become  indefinitely  small.  Then  if  a  unique 
limit  is  approached  by  the  sum  regardless  of 
the  way  in  which  the  subdivision  of  ab  is  made, 
we  have 

lim  tifihi=  I    f{x)dx  . 

I  */  a 

Necessary  and  sufficient  conditions  for  the 
existence  of  this  limit  are  well  known.  In 
particular,  if  the  function  j{oc)  is  continuous 
over  the  interval  ah  or  has  at  most  a  finite 
nmnber  of  discontinuities,  the  limit  and  hence 
also  the  integral  exists. 

2.  Now  let  us  form  the  product 

a  ■  h    c . 

This  operation  is  associative  and  commutative, 
that  is  to  say, 

{ah)  c  ^^  a  [he) 
and 

ah  "=  ha  . 

It  is  not  worth  om*  while  to  consider  the  oper- 
ation which  could  be  obtained  from  a  product 
by  a  limiting  process  such  as  the  one  employed 


LECTURE   I 


in  defining  an  integral.     We  should  be  led  to 
logarithmic  integration. 

3.  However,  let  us  consider  a  limiting  pro- 
cess which  leads  us  to  something  more  than 
these  elementary  operations. 


Let  us  choose  a  set  of  numbers 


m, 


where 


i,s^ 

-1,2, 

•    9 

an  array 

mn 

mi2 

w^2l 

^22 

Mg^ 

nigz 

and 

numbers 

nis. 

that 

is, 

nn 

nn 

n2i 

^22 

which  may  be  written  in 


m 


Iff 


7)h 


Ig 


m 


99 


where  /,  6=  1,  2, 


9 


*ffi 


'ff2 


''^9 


^20 


We  then  consider  the  operation 

9 

(1)     S„  riUj,  nf,^ 
1 

which  we  shall  call  composition  of  the  second 
type.     This  operation  is  associative,  for  if  we 


6  THEORY  OF  PERM  LIABLE  EL'XCTIOXS 

also  introduce   a   set   of  numbers  jhsi  where 

2,  s  =  1,  2      .     .      (J  ,  and  write  the  sum 

y      y 

X,,  %k  m^i,  n,,k  J)/,, 

1      1 

the  expression  which  we  thus  obtain  is  equiv- 
alent to  either  of  the  forms 

.'/       y 

S/,   (2;,  lUif,  n,„)  J),,, 

1        1 

9  y 

2/,  ntif,  (2^.  ti;,/,  j)ks) 
1  1 

which  f)i'oves  that  the  associative  law  is  sat- 
isfied. 

INIaking  use  of  the  notation 

1 
we  shall  have 

which  may  be  written  without  the  parenthesis, 
thus, 

The  commutative  law  will  in  general  not 
be  satisfied.  When  it  is,  the  quantities  under 
consideration  are  called  permutable,  and  we 
have 


LECTURE    I 


{m,  n)i,  =  {n,  m)i,.  . 

We  can  at  once  give  an  example  involving 
permutable  quantities.  All  that  is  necessary 
is  to  consider 

{m,  m)ir  which  nia}^  be  written    (m^)ir 
[m,  m,  m)ir  which  may  be  written    {ni^)ir 
and  so  on.     And  it  is  clear  that 

since  the  associative  law  is  satisfied. 

4.  We  shall  consider  also  another  operation 
similar  to  the  last,  namely 


(21 


s— 1 


which  will  be  called  composition  of  the  first 
type.  The  smn  (1)  previously  considered  re- 
duces to  this  one  if  we  suppose  that  the  num- 
bers are  zero  unless  the  second  subscript  is 
greater  than  the  first.  In  other  words,  we 
have  in  this  case 
0 


muy  m^ 


n-i 


0     0 


7)1 


23 


0     0      0 
0     0     0 


m 


f/-i,y 


8  THEORY  OF  PERMUTABLE  FUNCTIONS 

Let  US  represent  the  sum  (2)  by 

This  expression  vanishes  if  s  is  less  than  or 
equal  to  /  +  1.    Moreover,  if  we  write 

we  shall  have 

which  vanishes  if  s  is  less  than  or  equal  to 
I  +  2,  and  so  on. 

In  general,  it  is  not  true  that 

but  when  this  condition  is  satisfied,  the  two 
quantities  are  called  permutable.  To  distin- 
guish this  sort  of  permutability  from  that 
which  we  defined  in  section  3,  we  shall  say  that 
the  new  and  the  old  are  of  types  one  and  two 
respectively.     In  other  words,  if 

g  g 

(8)    S;,  ;;/,;,  nj,g  =  2,,  n^,  m,„j 
1  1 

we   have   permutability   of   tlie   second   type, 

whereas  if 

s-l  s—\ 

i+l  i+1 


LECTURE   I  9 

we  have  permutability  of  the  first  type. 
Clearly,  if  we  put 

we  at  once  obtain  an  example  of  permutability 
of  the  first  type  like  the  one  mentioned  in  the 
last  section.  Nevertheless,  we  shall  give  an- 
other example  which  is  of  interest.  Let  us 
suppose  that  n^^  =  1.  Then  the  condition  for 
permutability  becomes 

.s— 1  S-1 

Putting  s  =^  I  -\-  2,  \\Q  have 

and  putting  .s  ^  ^  +  3,  we  have 

whence 

and  so  on.    Owing  to  the  above,  we  have 

for  all  values  of  r,  s,  and  g.  From  this  it  fol- 
lows that  the  matrix  of  the  ins  is  of  the  follow- 
ing type: 


10 


THEORY  OF  PERMUTABLE  FUNCTIONS 


0     ^^1    i(.>    ((.^ 

U  U  ((i  (ir, 

0     U     U     f^i 


'y-1 


^g-i 


U     0    u     u    . 

U    0    0    0    .    .    .    u . 
The  law  for  this  matrix  may  he  expressed  hy 

Tih,  =  ;«,_,• , 
which  puts  into  evidence  the  fact  that  the  val- 
ue of  m     depends  upon  the  difference  between 
the  two  subscripts  8  and  i. 

5.  Now  what  do  we  find  when  we  pass  to 
the  limit  by  a  process  analogous  to  tlie  one 
employed  in  the  integral  calculus  in  going 
from  a  sum  to  an  integral?     We  there  passed 

from  a  set  of  quantities  /i,  f-^,  /a,    f,, 

with  single  subscripts  to  a  function  f{d')  of 
one  independent  variable  aj,  the  variable  taking 
tlie  place  of  the  subscripts.  Here,  we  have  in- 
stead a  set  of  quantities  ;«,.^  with  double  sub- 
scripts; hence,  we  must  replace  them  by  a 
function  of  two  variables 


/G^,.y) 


LECTURE   I  11 


where  the  two  variables  take  the  place  of  the 
double  subscripts.  Moreover,  we  also  have 
another  set  of  quantities  „  .^  which  must  be  re- 
placed by  a  different  function  of  two  variables 
(l>{:v,^).    Finally,  2/1  %7i  W;,s  must  be  replaced  by 

J/(.r,  0  (/>(£,  ^)r/£. 
We  thus  obtain  two  operations: 
Composition  of  the  first  type: 

Composition  of  the  second  type: 

^    a 

The  condition  for  permutability  of  the  first 
type  is 

ff[x,  ^)  <^(e,  //)  d^  =f'^(x,  i)  M,  y)  dt. 

for  permutabilitj"  of  the  second  type, 

ff{x,^.)<^{^,U)d^=Pi>[x,^)f[^,y)  d^^  . 

a  a 

The  associative  property  is  always  satisfied. 
6.  Let    us   begin   by   examining   permuta- 


12  THEORY  OF  PERMUTABLE  FUNCTIONS 

bility  of  the  first  type.  The  most  miportant 
facts  are  here  summarized: 

1.  All  of  the  functions  which  can  be  ob- 
tained by  the  composition  of  permutable  func- 
tions are  permutable  with  one  another  and 
also  with  the  original  functions. 

2.  All  of  the  functions  which  can  be  ob- 
tained by  the  addition  or  the  subtraction  of 
permutable  functions  are  permutable  with  one 
another  and  with  the  original  functions. 

Now,  let  us  see  how  the  following  problem 
may  be  solved:  To  determine  all  the  func- 
tions whicli  are  permutable  with  unity. 

We  can  readily  solve  this  problem  if  we  re- 
call a  question  whicli  has  already  been 
answered.  For  before  passing  to  the  limit,  we 
saw  that  if  the  functions  ?« .,  were  permutable 
with  unity,  the  condition 

was  satisfied.  Now  since,  in  the  limit,  the  sub- 
scripts are  replaced  by  the  variables  07  and  y, 
we  are  led  to  infer  that 

This  we  can  prove  immediately.     For  if 


LECTURE   I  13 

^    X  X 

it  must  follow  that 

d  (h  3  6        .^       ^ 

Hence  ^  and  /  are  of  the  forms  ^{y — x)  and 
j[y — x)  respectively. 

Moreover,  all  of  the  fmictions  of  the  type 
/(?/ — x)  are  permutable  with  one  another;  for 

as  can  be  verified  at  once.  The  functions  of 
type  f(y — x)  form  a  group  of  permutable 
functions  which  is  of  especial  interest.  We 
have  called  it  the  group  of  closed  cycle.*  How- 
ever, we  shall  not  go  into  an  examination  of 
it  here. 

7.  We  have  used  several  different  notations 
representing  the  operation  of  composition. 
The  simplest  scheme  where  no  confusion  with 
multiplication  is  liable  to  arise,  is  merely  to 
write 

f(f>      or     f(f>ix,?/) 

*Le5ons  sur  les  equations  integrales  et  int6gro-differentielles. 
Paris:   Gauthier-Villars.     1913.     P.   150. 


14  THEORY  OF  PERMLTABLE  FUNCTIONS 

to  rej^resent  the  resultant  of  the  composition 
of  two  function  /  and  (f) .  But  in  a  case  where 
confusion  might  arise,  we  may  place  a  small 
star  over  the  letters,  thus 

f4.. 

We  may  also  put  the  letters  in  square  brackets 

[/■■^]- 

just  as  in  the  above  we  wrote  [;;/.  71']/,,,  to  rep- 
resent the  composition  of  the  quantities  niij^ 
and  mfj, . 

To  indicate  the  composition  of  /  with  itself, 
the  composition  of  the  resultant  thus  obtained 
with  f,  and  so  on,  we  shall  write 

r-,/',  .... 

respectively,  and  if  no  confusion  with  multi- 
plication is  liable  to  arise,  we  may  even  omit 
the  small  stars  and  write 

J      ■>    J      1 

8.  The  notation  in  certain  cases  demands 
particular  examination.  Thus,  to  indicate  the 
product  of  a  constant  a  by  /,  we  write  «/;  and 
if  h  is  also  a  constant  and  4)  a  function  which  is 


LECTURE   I  15 

permutable  with  f,  then  6  ^  is  permutable  with 
af  and  the  composition  of  the  two  gives  us 

«  » 
a  h  f  ^  . 

Moreover,  if  we  have  two  polynomials  made 
up  of  permutable  functions  with  constant  co- 
efficients, these  polynomials  will  also  be  pre- 
mutable  with  one  another,  and  to  effect  their 
composition  all  that  is  necessary  is  to  apply 
the  same  rule  which  is  used  when  polynomials 
are  multiplied  together. 

But  if  a  and  h  are  constants,  then  a  +  / 
and  &  +  0  will  not  in  general  be  permutable 
with  one  another.  Nevertheless,  we  shall  ex- 
tend the  definition  so  as  to  have  in  this  case 

(a+f)  (b-{-4)  =  ab  +  «(^  +  hf  +  f^  . 

9.  Before  going  further,  let  us  consider 
what  takes  place  in  the  case  of  composition 
of  the  second  type. 

All  that  we  have  said  above  concerning  per- 
mutability  of  the  first  type  can  be  established 
for  permutability  of  the  second  type  barring 
the  remarks  on  permutability  with  unity. 
With  this  exception,  all  of  the  properties  just 
mentioned  may  be  extended  to  this  case  at  once. 


16  THEORY  OF  PERMUTABLE  FUNCTIONS 

10.  Let  US  pass  in  review  some  of  the  most 
interesting  properties  which  can  be  derived 
from  the  operation  of  composition.  We 
shall  return  once  more  to  the  finite  case  and 
consider  the  operation  of  composition  for  the 
numbers  viig.  We  saw  above  that  if  we  com- 
posed the  w^s  with  the  n's,  the  resultant  thus 
obtained  with  the  p^'s  and  so  on,  then  after 
s-i-1  compositions,  the  resultant  would  be  zero. 
In  a  like  manner,  all  of  the  symbolic  powers 
of  7Jiig  beginning  with  [7}f~^^^~\i,  have  a  value 
zero. 

Having  seen  this,  let  us  consider  any  ana- 
lytic function 

00 
0 

which  converges  within  a  certain  circle,  and 
let  us  write 

0 

This  new  expression  is  evidently  an  integral 
rational  function  of  z,  that  is,  a  polynomial. 

Moreover,  this  result  may  be  generalized. 
Consider  the  analytic  function 


00  00 


0         0  0  ^ 


LECTURE  I  17 

of  more  than  one  variable,  and  write 

0        0  0 

*1       ^2        .     .     .    2g   <=  . 

We  again  obtain  a  polynomial. 

11.  Now,  what  does  the  above  theorem 
become  when  we  pass  by  a  limiting  process  to 
the  composition  of  functions?  This  we  pro- 
ceed to  investigate. 

Consider 

0 

We  shall  prove  that  if  f  (x,  y)  is  finite „  this 
eocpression  is  always  an  entire  function  of  z, 
whatever  may  he  the  absolute  value  o/  f  (x,  y) . 
To  prove  this  theorem,  we  notice  that 

where  M  is  some  finite  quantity  and  where  It 
is  less  than  the  radius  of  convergence  of  the 
series.  Moreover,  let  /u,  be  a  quantity  which 
is  larger  than  the  absolute  value  of  /.  Then 
we  shall  have 


18  THEORY  OF  PERMUTABLE  FUNCTIONS 

2! 

\J\^,i/)\< — 31 —  •  •  •  • 

and  so  on,  whence 

which  proves  that  the  series  is  convergent  for 
all  values  of  z  and  is  thus  an  entire  function 
of  %. 

This  theorem  may  also  be  generalized.    Let 
us  consider  the  series 

00  00 

Sh    .    .     .    Si,  y4i  .     i     Z^'    .     .    .    Z,'e 

0  0  ^  " 

which  represents  the  expansion  of  a  function 
F [z^,  Z2,  23, .  . .  .Ze)  about  a  point,  and  which 
converges  if  the  absolute  values  of  Zi,  Zo,  S3, 
.  .  .  .Zg  do  not  exceed  certain  limits.  Then 
the  series 

0  0^ 

is  always  an  entire  function  of  z^,  z^,  Z2,.  .  .s„. 


LECTURE   I  19 

The  proof  is  made  in  the  previous  case.  Thus, 
if  fi>  f-ij  fs,'  '  ■  -fe  are  each  less  than  /a  in  abso- 
lute value,  then 

uh+.--+ie  f^  —  M...  +»,_! 
fn  fi.\  <  —^ ^ ^ — 

l-^i  ••••^'''l^(z\+4+  ...  +4-i)i 

and  hence  the  theorem  may  be  verified  im- 
mediately. We  may  also  demonstrate  another 
property  besides  the  one  just  shown.  Indeed, 
we  have  up  to  the  present  regarded  the  func- 
tion F (zi,.  .  .  .z^,  \  oj,  y)  as  Si  function  of  Zi,  Z2, 
Zs,  S4,  ....  Zp,  but  it  is  also  a  f miction  of  cc 
and  y.  Regarded  as  a  function  of  these  two 
variables,  the  function  is  permutable  with  the 
functions  /i  ....  fe.  This  may  be  seen  at 
once;  for  owing  to  the  uniform  convergence 
of  the  series,  the  operation  of  composition 
may  be  performed  term  by  term,  and  since 
each  term  of  the  series  is  permutable  with  /i, 
/2>  fs,  ..../(.,  so  also  must  be  the  sum. 

To  sum  up,  we  have  the  following  theorem: 

If 


22  .  .  .  XAi 


i^    *1 


n 


is  the  expansion  of  aii  analytic  function  about 
a  point,  then 


20  THEORY  OF  PERMUTABLE  FUNXTIONS 

where  f i,  io,  f 3, .  •  .  .  f e  are  permutable  func- 
tions, is  an  entire  function  of  Zi,  z^,  Z3, .  .  .  .Ze, 
aw6^  a*  a  function  of  x  aw^Z  y  is  ijermutahle 
with  the  functions  fi,  £2,  fa  ....  fe. 

Now  if  in  F  [z^,  z^,.  . .  .z^  \  oc,  y)  we  put 
Si  =  2^2  =  ~3  =  ....  Sg  =  1,  we  obtain  a  series 

which  is  convergent  for  all  values  of  the  fs. 

12.  The  theorems  which  we  have  been  de- 
riving above  suggest  a  method  for  investigat- 
ing to  a  considerable  extent  the  j^roperties  of 
permutable  functions  and  for  carrying  out  the 
operations  of  composition. 

Thus,  let  us  consider  any  analytic  expression 

F  {,,...,,) 

which  can  be  expanded  about  the  point  z^  = 
^2  —  S3  =  2:4  =  ....  Ze  =  0  in  powers  of  Zi, 
Zo,  z-i,  .  .  .  .  Zg.  If  we  replace  z^,  Zo,  z^,  ....  Zg 
in  the  series  by  /i,  /2,  /s,  •  •  •  •  /.  respectively, 
and  write  the  operation  of  composition  wher- 
ever we  previously  had  multiplication,  we  shall 


LECTURE   I  31 

always  obtain  a  series  which  converges  for  all 
values  of  /i,  /a,  fs,  ....  fe,  and  which  repre- 
sents a  function  permutable  with  fi,  f2,  f-i,  .... 
fe .    We  may  represent  it  by 

Thus,  every  algebraic  expression  takes  on  a 
new  meaning  for  the  operation  of  composition. 
For  example, 

— y        ..,2  _l_  ^3 

is  a  series  which  converges  within  the  unit  cir- 
cle.    But  if  we  write 

J  •       •        • 

1  +  /  -^  ^  '' 
we  obtain  a  series  which  converges  for  all 
values  of  /  and  which  is  permutable  with  /. 
Consequently,  a  meaning  has  been  ascribed  to 
the  expression  on  the  left  hand  side  of  the 
equation. 

Moreover,  if  we  take  two  expressions 


and 


22  THEORY  OF  PERMUTABLE  FUNCTIONS 

~, i.=  /  +  /'+/'+    •  .  . 

1-/ 

then  to  make  the  composition  of  the  two  left- 
hand  members,  it  is  only  necessary  to  apply 
the  rules  for  finding  their  algebraic  product 
and  we  shall  have 

.  =  /■2+/^+  . . . 
i-f     ^     ^ 

Hence,  all  the  rules  of  ordinary  algebra  re- 
main valid  when  we  pass  from  the  field  of  mul- 
tiplication to  the  field  of  composition. 

Some  of  the  consequences  which  can  be  de- 
rived from  this  fact  will  be  seen  shortly. 

13.  Now  let  us   see  what  takes  place   for 
the  second  type  of  composition. 
Let 

be  the  ratio  of  two  entire  functions  ^{p^)  and 
1  +  i//  (2;)  which  are  such  that  </)(0)  =  v/'(0) 
=  0. 

Then  we  shall  have  an  expansion 


LECTURE  I  S3 

F{z)  =  A,s  +  A.,z^  +  A^z^+  ... 
which  converges  in  general  within  a  certain 
circle  having  the  point  ;:;  =  0  as  center. 
Now  let  us  consider  the  expression 
FU^-)  =  A,m,,z  +  A,(m%z^+... 
We  shall  prove  that  this  new  series  is  the  quo- 
tient of  two  entire  functions. 

14.  Let  us  first  write 

<t>(z)  =  B,z  +  Bo  .r  +  .  .  . 
and  determine  a  quantit}^ 

^is{^)  =  ^1  ^^hs  •«•  +  B2(m%  z^+  .  .  . 
We  say  that  ^j,  (z)    is  an  entire  function  of  z. 
For  let  ju,  be  greater  in  absolute  value  than  m^s . 
We  then  have  (§3) 

I  w^,,|  <(i,    I  {m'%  \<g  li\    \  {ni^)is  !</>%... 
Moreover, 

1^1  \iiz  +    \B,  \^'gz'  +    1^3  \^r^+  ••• 
converges  for  all  values  of  z,  and  the  theorem 
is  proved. 

For  the  same  reason,  if 

.\,(,)  =  C,,+  C,^+  ... 
then  the  series 

x\fi,{^)  =  C^  mt,  .*•  -V  C\{7n%,  £^  +  ... 
is  an  entire  fmiction. 


24  THEORY  OF  PERMUTABLE  FUNCTIONS 

Bearing  these  facts  in  mind,  let  us  consider 
the  system  of  algebraic  linear  equations 

g 

1 
If  we  replace  the  unknowns  X,^  by  F^^  we 
can  verify  without  trouble  that  these  equa- 
tions are  identically  satisfied.  But  if  we  solve 
for  the  unknowns  X,,  in  the  above  system,  the 
solution  will  be  expressed  as  the  quotients 
of  rational  entire  functions  of  i//,^  and  <^j,  and 
hence  the  quantities  X^^  are  quotients  of  entire 
functions  of  z. 

It  is  clear  that  the  determinant  which  con- 
stitutes the  denominator  of  these  quotients 
cannot  vanish  identically,  and  hence  the  theo- 
rem is  2)roved. 

It  is  not  difficult  to  generalize  this.  Thus 
instead  of  the  quotient   (•!),  let  us  write 

where   (f)    and   xjj   are  entire  functions  of  the 
variables  Zi,  Z2,  Zs,   ....   Zg  which  vanish  for 

Zi         Z2        S3  ,  .  .  .  Zc         U.      xl 

is  the  expansion  of  F  about  the  point  for  which 


LECTURE  I  25 

Zi  =  Z2  =  ^z  =  ....  =  Ze  =  0,  and  if  we  write 

F,,{z,  ,...z,)  =  t..  .tA,^ . . .  ,^(nf^n'"- . . .  q'^)  z^\  .  .^/a 

where  m  ,  n  ,  ....  q  are  permutable,  then 
the  function  F  (si,  S2,  S3,  .  .  .  .  s^  |  .r,  ?/)  will  be 
the  quotient  of  two  entire  functions  of  Z\,  z^, 

Zz,    ....    Ze. 

To  make  the  proof  in  this  case,  it  is  only 
necessary  to  repeat  the  argument  given  above. 
We  may  add  that  Fi^  is  permutable  with  m^, 

"is  J  •  •  •  •  ^ts   • 

15.  Let  us  now  pass  to  the  limit,  that  is, 
let  us  consider  permutable  functions  of  the 
second  type*.  All  of  the  theorems  remain 
valid.  In  other  words,  we  have  the  theorem 
that 
F(,,,  ...^  x,y)  =  t...  tA,^ . .  ,;^/V> . .  .f:^  zt .  • .  ^;', 

where  j\,  J2,  jz^  -  -  -  -  /«  are  permutable  func- 
tions of  the  second  type,  is  the  quotient  of  two 
entire  functions.     Also, 

F[z^,...z,\x,y) 

*  To  indicate  composition  of  the  second  type,  we  shall  make 
use  of  a  double  star  thus: 

•*     ** 
/l    1\ 


26  THEORY  OF  PERMUTABLE  FUNCTIONS 

is  a  function  which  is  permutable  with  fi,  f2, 

U,    ■■■■    fe. 

We  shall  study  some  applications  of  these 
fundamental  theorems  in  the  second  lecture. 


LECTURE  II 


LECTURE  II 

1.  We  shall  begin  by  classifying  integral 
equations  into  several  categories.  First,  let 
us  examine  those  which  are  linear.  The  sim- 
plest ones  which  we  rmi  across  are  the 
following : 

(1)     /  W  +  /oVW  F[x,y)  dx  =  ^y)  , 

known  as  Volterras  equation  of  the  second 
kmd,  and 

(10    f{y)-^§\f(x)F{x,y)dx  =  <l>(y), 

Fredholm's  equation  of  the  second  hind. 
We  shall  also  consider  certain  other  kinds  fur- 
ther on. 

Let  us  look  at  equation  (1).  If  we  multi- 
ply both  sides  by  ^  (y,  z)  and  integrate  with 
respect  to  y  between  the  limits  0  and  z,  we 
obtain 

jl^(y,z)fiy)dy-^  ^]f(x)dx  jy(x,y)^(y,z)dy 

=  ^\^[jj)^{y.^)dy . 


30  THEORY  OF  PERMUTABLE  FUNCTIONS 

If  now  the  function  ^  be  so  chosen  that 

(A)  j[F{x,i/)  $ [i/,  z)  dy  =  -  <^[x, ,)  -  F(x,  z), 
it  will  follow  that 

/  0  ^  (^'  ^)  f^y)  '^y~  ^\  *(-^^'  ^)  /(^)  ^^^ 

-  /J  F{x,  z)  f{x)  dx  =  f'jiy)  <D(y,  z)  dy 
which  is  reducible  by   (1)   to 

/(.•)  =  <^(.~)+/V(^)(D(^,,.)./y, 

^  0 

so  that  the  difficulty  has  been  narrowed  dowTi 
to  the  solution  of  (A).  In  the  symbols  for 
composition  of  the  first  kind,  this  equation 
maj^  be  written  as  follows: 

(2)     ^{x.y)-^  F{x,y)-\-  F6{x,y)  =  ^  . 
If  we  apply  a  similar  argument  to  equation 
(1')  we  find  that  the  solution  will  be  given  in 
the  form 

f{z)=^{z)^  f^{y)^[y,z)dy 
^  0 

where 

(2^)     ^{x,y)  +  F(x,y)  +  F*i(x,y)  =  0  . 

2.  Let  us  first  see  how  equation    (2)   may 
be  solved.     If  we  write 


LECTURE  II 

z 

?  +  2"!  +  Z  Zi  — 

0 

we  shall  obtain 

—  z 

=  -.  +  /- 

_  -3 

<5 

+ 

"^~1  +  ^ 

the  solution  heir 

[g  valid  if     z 

< 

1. 

But  suppose 

we  write 

_      ^     _ 

=  -F+  F^- 

.  j^3 

+ 

31 


1  +  F 
Then,  in  this  case,  we  have  an  expansion  which 
converges  for  all  values  of  F  and  which  satis- 
fies equation    (2)    by  what  we  have  proved. 
Hence  we  shall  have 

$=  -F+  F''-F^+  .  .  . 
and  the  integral  equation  (2)  is  solved. 

If  we  replace  F  (oD,y)  hj  u  F  (x,y)  in  equa- 
tion  (2)  we  obtain 

^{x,f/)  +  u  F(x,fj)  +  u  Fi{x,/j)  =  0 

^=  -uF+u'^F-'-n''F^+  .  .  . 
which  series  is  always  an  entire  function  of  u. 

3.  Turning  to  equation  (2'),  let  us  replace 
F  by  uF  as  in  the  previous  case.  We  shall 
then  have 


32  THEORY  OF  PERMUTABLE  FUNCTIONS 

^  +  uF  +  r(F$  =  0, 
and  as  a  consequence  of  the  last  theorem  of 
the  first  lecture,  we  have  that  ^  can  be  ex- 
pressed as  the  quotient  of  two  entire  functions 
of  u. 

4.  As  soon  as  we  have  stated  the  fun- 
damental problems  in  this  form,  it  is  easy  to 
see  that  they  are  only  special  cases  of  other 
classes  of  problems  of  a  much  more  general 
nature. 

Indeed,  let  us  consider  any  analytic  func- 
tion F{z-i,  Z2,  Zs,  ....  s„)  whatsoever  and  write 
the  equation 

(.3)     F{,„z,,^„....z„)=0. 
Furthermore,  let  us  suppose  that  this  equation 

is  satisfied  hy  Zi  =  Z2  =  ^s  =■•■■  —  ^n  =  0>  and 
let  us  regard  z„  as  a  function  dependent  upon 
2i»  ^2,  Z3,  ....  Zn-\'  If  the  point  Si  =  So  —  S3  — 
....  =  Zn  =  0  is  not  a  critical  point,  we  may  de- 
velop 3„  as  a  power  series  in  Si,  S2,  •  •  •  •  ^n-i  and 
the  expansion  will  be  convergent  within  some 
region.     We  shall  thus  have 

(4)     ^„  =  S  2...  S^, ■...,;__.-/....  4-1', 

-^00.  .  .0  ^^  '-'  • 


LECTURE  II  33 

Now  suppose  we  replace  Zi,  Z2,  .  .  .  .  s«  in  equa- 
tion (3)  by  the  permutable  functions  fi,  f^,  .  . 
....  /  „  respectively  and  regard  the  operations 
as  compositions  of  the  first  type.  Then,  in 
terms  of  our  notation,  we  shall  have 

^(/,  A    .  .  .   fn)  =  0    . 

The  equation  which  we  have  just  found  will 
no  longer  be  algebraic  or  transcendental  but 
will  be  an  integral  equation,  since  the  oper- 
ation of  composition  is  an  operation  of  integra- 
tion. Nor  will  the  equation  in  general  be 
linear  as  was  equation  ( 2 ) ,  but  of  any  degree 
whatsoever.  Nevertheless,  if  we  regard  /„  as 
the  unknown  function,  we  shall  be  able  to  find 
its  solution  by  the  same  process  which  we  used 
in  solving  equation  (3).  Indeed,  it  is  only 
necessary  to  replace  Si,  Z2,  %,  ....  s„  in  the 
series  (4)  by  fi,  fz,  fz,  •  •  •  •  /„  respectively  and 
to  treat  the  operations  as  operations  of  com- 
position.    In  this  manner,  we  find 

(5)    /,.  =  2  2...2.1,,...,„_,/,'.  ...ylV. 

An  interesting  fact  to  be  noticed  is  that 
whereas  the  expansion  (4)  is  in  general  con- 


34  THEORY  OF  PERMUIABLE  FUNCTIONS 

vergent  over  a  limited  region  only,  the  solution 
(5)  is  valid  for  all  values  of  fi,  f2,  ....  fn-\- 
Evidently  problems  of  integration  are  of  a 
more  complicated  nature  than  algebraic  or 
transcendental  problems,  yet  we  have  the  sur- 
prising and  interesting  result  that  the  solu- 
tions of  the  former  are  much  more  simple  in 
the  sense  that  the  regions  over  which  they  are 
valid  is  infinite. 

We  may  also  replace  Zi,  Z2,  -3,  ....  s„_i 
in  equation  (3)  by  ^1/,  S2/2, s„_i/„_i  res- 
pectively and  write  the  equation 

Then  the  solution  will  be 

Jn  ^   '  '  '    ^  ^  i,  i        ,     ~1     •   •  •    -^n-l    J\     •  •  •/n—\ 

1   •    •   •      ?! — I 

which  is  of  the  form 

/»(.e-i,...2'„_i|.r,  y)  . 
The  series  will  be  an  entire  function  of  Zu  ^2, 
Z3,   ....   s„_i ,  and  with  respect  to  tV  and  y, 
f  n  {^1,  22;,  ^z,-  •  '  .^n-i  \  ^^y)  will  be  permutable 
with  the  functions  /,,  /2,  /,i,  ....  /„_i . 

5.  AVe  might  also  start  with  a  system  of 
equations,  as 


LECTURE  II 


35 


(6)      i 


which  are  satisfied  when  ;:;i  =  Ui  =  Zo  —  U2  — 
.  .  .  .  =  z,,  =  0.  Now  let  us  suppose  that  we 
can  define 


f  %=s  2...  2^; '..,,._  .-/■..•<» 
{')   i  

Im,  =  XS...S.4'«       ,    .?,''...^'.. 

as  imj)Hcit  functions  of  Zi,  Z2,  S3,  .  •  •  •  z,,  which 
have  no  critical  point  at  Zi  =  Z2  —  .  .  .  .  Zn  =0. 
Then  if  we  write  the  integral  equations 

*  *     *  *  . 


F^  (*1  /i,  . .  .  *„  /„,  <^i,  .  .  .  ^p)  =  0 

where  fi,  f2,  fz,  •  •  •  •  fn  are  permutable  func- 
tions, the  solution  of  the  system  will  be 

fh    =    ^  ■<     J  (1)  ^  h  ~  'n     fn  f\i 


<^,,  =  S...S^lf 


n      fti  f'n 

n        J  \     •  ••J  n 


36  THEORY  OF  PERMUTABLE  FUNCTIONS 

and  the  functions  thus  obtained  will  be  entire 
functions  of  Zi,  z^,  Zz,  ....  s„  for  all  values  of 
/i>  fi,  fz,  ■  ■  fw  Moreover,  these  solutions 
will  be  permutable  with  the  given  functions. 
All  of  the  equations  which  we  have  been  con- 
sidering involve  only  integrals  for  which  the 
limits  of  integration  are  x  and  y;  that  is  to 
say,  they  are  equations  with  variable  limits. 
Let  us  see  what  the  situation  is  when  the  limits 
are  constant.  Returning  to  the  set  of  equa- 
tions (6),  we  shall  suppose  that  the  solutions 
(7)  are  quotients  of  entire  f mictions.  Then 
let  us  examine  the  integral  equations 


-^p(^l  flj-"   -nfn,    ^l,---<t>p)  =  ^ 

where  /i,  fo,  fz,  •  -  -  -  f  n  are  permutable  func- 
tions of  the  second  type.  In  these  equations 
the  limits  of  integration  are  constant,  since  we 
are  concerned  with  composition  of  the  second 
type. 


LECTURE  II 


37 


If  now  we  put 


"1 


y.'^ 


0p=2: 


the  functions  thus  obtained 

1)  satisfy  the  preceding  system  of  equa- 
tions, 

2)  are  quotients  of  entire  functions,  and 

3)  have  permutabihty  of  the  second  type 
with  the  original  functions  fi,f2,  •  •  •  •  /n  • 

Thus  we  see  that  hnear  equations  are  only 
a  very  special  type  of  integral  equations  and 
that  we  can  pass  from  their  study  to  that  of  a 
more  general  class. 

6.  Let  us  prove  certain  important  proper- 
ties about  functions  which  may  be  found  by 
a  process  like  the  one  above  outlined.  More 
precisely,  let  us  show  what  certain  algebraic 
properties  become  when  we  pass  from  multi- 
plication to  composition.  We  shall  begin  by 
giving  an  example: 

We  consider  the  exponential  function 


38  THEORY  OF  PERMUTABLE  FUNCTIONS 

Z^  z  ^ 

Associated  with  it  is  an  addition  theorem 

g(2  +  2l)     =    qZ   g2l     ^ 

Suppose  we  put 

AVe  then  have 

(8)    /(.-  +  ^i)  =/(.a')/(.=-i)  +/(.0  +/(-'.)  • 
Keeping  the  above  in  mind,  let  us  write  the 
function 

-.2  r-2.  .,3  r3 

We  can  see  at  once  what  the  relation  (8) 
becomes.  Indeed,  we  have  only  to  replace 
multiplication  by  comj^osition.  We  shall 
therefore  have 

V(z  +  ,,  I X,  y)  =  V{,  I  :v,  ij)  +  V  (.ci  I X,  //) 

+  V{z\x,y)  V[s^\x,y)  , 
that  is 

V{,  +  z;  1 X,  y)  =  V  (..  I X,  y)  +  V  (.z,  \  x,  y) 

+  f  V(z\x,  ^)  V{,,\^,  y)  d^  . 

X 

In  other  words,  the  theorem  of  algebraic  ad- 


LECTURE  II  39 

dition  for  the  exponential  function  becomes  for 
this  new  fmiction  a  theorem  of  integral  addi- 
tion as  we  have  called  it.* 

7.  To  go  from  the  particular  case  to  the 
general  involves  no  difficulty.  Consequently, 
we  may  state  the  theorem:  To  every  theorem 
of  algebraic  addition,  there  corresponds  a  the- 
orem of  integral  addition. 

Thus,  for  example,  if  we  consider  elliptic 
functions,  we  can  pass  from  these  to  entire 
functions  by  the  process  of  §  11  of  the 
preceding  lecture.  To  the  addition  theorems 
for  elliptic  functions,  there  correspond  new 
addition  theorems  for  new  functions.  In  a  like 
manner,  let  us  consider  the  <j  function  of 
Weierstrass.  Suppose  we  examine  for  a  mo- 
ment the  expansion  of  this  function  and  replace 
u  in  the  expression  by  uf(iT,  y)  where  the 
powers  of  /  represent  operations  of  composi- 
tion. The  three-term  equation  for  cr  leads  us 
to  a  three-term  equation  for  the  new  function 

*  Evans  has  studied  in  a  systematic  manner  an  Algebra  of 
permutable  functions.  (Memorie  Lincei,  S.V.  Vol.  VIII, 
1911;  also  Rendiconti  del  Circolo  di  Palermo,  Vol.  XXXIV, 
1912.) 


40  THEORY  OF  PERMUTABLE  FUNCTIONS 

which  is  of  the  integral  type  since  we  have 
replaced  products  by  compositions. 

8.  What  we  have  said  about  compositions 
of  the  first  type  may  be  repeated  for  composi- 
tions of  the  second.  Returning  once  more  to 
the  example  involving  the  exponential  func- 
tion, let  us  put 

•*  ** 

jf(.~|.r,^)=./+Yr+ 3r  +  ■•• 

This  function  is  also  an  entire  function  and 
we  shall  have 

+  W{z\cc,i/)  W{,,\x,y), 
or  in  other  words, 

a 

It  is  hardly  necessary  to  prove  that  the  above 
is  true  in  the  general  case. 

9.  A  similar  theorem  may  also  be  stated 
for  the  case  of  more  than  one  variable;  hence, 
all  the  theorems  about  Abelian  functions  may 
be  carried  over  to  the  domain  of  integration  by 
a  process  like  the  one  which  we  have  indicated. 


LECTURE  II  41 

10.  Let  US  now  return  to  linear  integral 
equations.     As  indicated,  equations    (1)    and 
(1')  are  of  the  second  kind.    Those  of  the  first 
kind  of  the  Volterra  and  Fredliolm  types  re- 
spectively may  be  written 

(9)  f  f(^^)n^,.^)dx  =  <!>{?/), 

^  0 

(90     f /(:l-)  F(x,y)  ch  =  cl>(^)  . 
^  0 

Leaving  out  of  consideration  equations  (9') 
which  can  only  be  attacked  by  methods  of  a 
different  sort,  let  us  consider  equations  (9). 
The  latter  may  be  reduced  to  equations  of  the 
second  kind.  For  we  can  differentiate  and 
obtain 

*^o  dy  ay 

If  F  [y,  y)   does  not  vanish,  we  can  divide 

by  F  [y^  y)  and  get  an  equation  of  the  second 

kind. 

J.i  F  (y,   y)    vanishes   identically,   the   last 

equation  is  still  of  the  first  kind.    But  if 

idF{x,y)\ 
\     dy      K=y 
is  not  zero,  then  by  a  second  differentiation,  we 


42  THEORY  OF  PERMUTABLE  FUNCTIONS 

shall  get  an  equation  of  the  second  kind,  and 
SO  on. 

li  F{oc,y)  is  such  that  F  (od,oo)  >  0,  we  shall 
call  it  a  function  of  the  first  order.    If  F  [x,  x) 

=  0  and  I  —  I  JO  ,  we  shall  call  it  a  function  of 

the  second  order,  and  so  on.  Hence,  if  the 
order  of  the  function  F  [x,  y)  in  equation  (9) 
is  determinate,  the  equation  can  always  be  re- 
duced to  one  of  the  second  kind  by  a  finite 
number  of  differentiations,  and  hence  can  be 
solved  by  the  method  which  we  have  indicated. 

But  the  order  oi  F  (x,  y)  may  not  be  deter- 
minate. A  case  in  point  is  where  F  [x,x)  is  in 
general  different  from  zero  but  vanishes  for 
certain  values  of  x.  Then  the  nature  of  the 
problem  changes,  and  to  solve  it,  new  methods 
must  be  used.  To  develope  these  would  lead 
us  too  far  afield.  The  solution  of  this  question 
has  been  the  goal  of  numerous  enquiries.  We 
were  the  first  to  take  up  the  matter  and  since 
then  Lalesco  and  others  have  studied  it.* 

Instead  of  considering  equation   (9)   which 

*See:  Lalesco,  Introduction  a  la  thforie  des  equations  int^- 
grales.    Paris:  Hermann,  1912.     Troisifeme  partie  I. 


LECTURE  II  43 

is  of  the  first  kind,  we  may  consider  the 
equation 

where  we  can  regard  F  and  i//  as  the  known 
functions  and  <l>  as  unknown.  For  we  have 
only  to  suppose  that  aj  is  a  constant,  when  the 
equation  reduces  at  once  to  equation  (9) . 

If  we  take  the  equation  of  the  first  kind  in 
this  form,  we  may  also  write  it 

that  is  to  say,  the  problem  is  of  the  following 
nature:  Given  a  function  i//  which  is  the  re- 
sultant of  the  composition  of  F  and  <&,  and 
given  one  of  the  factors  F  of  the  composition, 
to  find  the  other  factor  <I>.  If  for  the  moment 
we  were  to  replace  the  operation  of  composi- 
tion by  that  of  multiplication,  the  problem 
would  reduce  to  that  of  finding  the  inverse 
operation;  that  is,  we  are  dealing  with  a 
problem  which  is  analogous  to  the  problem  of 
division. 

Now  it  is  necessary  to  observe  that  certain 
conditions  must  be  satisfied  if  the  problem  is 


44  THEORY  OF  PERMUTABI.E  FUNCTIONS 

to  have  finite  solutions.  Tlie  order  of  xjj  must 
be  greater  than  the  order  of  F  by  at  least 
unity.  For  when  oc=^y,  xjj  vanishes  to  a  higher 
order  than  F.  If  F  is  of  order  771  and  \jj  is  of 
order  n,  then  <l>  must  be  of  order  m-n.  More- 
over, two  cases  may  arise  according  as  the 
functions  F  and  xjj  are  or  are  not  permutable 
with  one  another.  Clearly  in  the  latter  case, 
^  cannot  be  permutable  with  F,  otherwise  the 
resultant  of  the  composition  of  the  two  would 
be  permutable  with  either.  But  if  F  and  i// 
are  permutable,  will  <I>  be  permutable  with  F 
and   i//  ? 

We  shall  prove  that  this  property  is  actually 
realized.     In  fact,  we  have 


*    *        *    • 


F^F  =  xfjF,         FF^  =  FxIj. 
Hence 

F{^F)  =  F{F^), 

and  since  this  integral  equation  has  but  one 
solution, 

^  F=  F^ , 
and  the  theorem  is  proved. 

11.  Furthermore,    when    the    problem    of 


LECTURE  II  45 

linear  integral  equations  of  the  first  kind  has 
been  put  in  the  form 

/  i  =  i|; 
other  problems   suggest   themselves   at   once. 
Thus,  if  F,  <i>  and  i//  are  known  functions,  we 
may  set  the  problem  of  determining  a  quantity 
such  that 

(10)  F  X+  X^  =  i/;; 

or  again  the  following  problem:  given  the 
known  quantities  .F:^,  Fo,  F^,  F^,  and  i//  to 
calculate  a  quantity  $  such  that 

(11)  >,  i  +  i  >3  +  #3  i  i^4  =  i/;  . 

The  above  are  new  equations  which  up  to  the 
present  have  never  been  studied  and  with  which 
we  shall  now  concern  ourselves. 

First,  let  us  consider  equation  (10)  which 
we  can  write 

Let  us  put 

X 

-fx{z;^)^{^,y)d^. 


46  THEORY  OF  PERMUTABLE  FUNCTIONS 

Then  we  shall  have 

g^  =  ~F{x, X)  X{x,  II)  +/^ F,{x.  I)  X(^,^)^£, 

^=  -^{U^U)  X{x,?/)  -  J  A%r,|)  ^,(^,y)  <Ii+'^  , 
where  we  have  put 

dF  a^ 

'  dx  ^  ^        d  Jl 

From  the  first  equation,  we  derive 
and  from  the  second 

+  jy{L^/)^-^^d^+ii{x,j/). 

where  /  and  ^  are  two  known  functions  which 
one  can  obtain  from  F  and  <I>  and  where 

is  also  a  known  function.  Then  by  subtracting 
tlie  second  equation  from  the  first,  we  have  at 
once 


LECTURE  II  4T 


dS{x,y)  _       1        dS{x,?/)  _  jjf^^^  ^^^ 


^il/^U)       ^//  F{x,x)       dx 

-\-  J  nx,^)-^Y  d^ 

-J  ^.W'-^^^;^  ^' 

and  integration  by  parts  gives 

Thus  we  are  led  to  the  following  result :  To 
solve  the  integral  equation  (10)  we  must  solve 
the  problem  which  presents  itself  in  the  shape 
of  the  last  equation.  This  problem  is  nothing 
more  than  the  integration  of  an  integro-diff  er- 
ential  equation.  Indeed,  equation  (12)  is  both 
of  the  type  of  an  integral  equation  and  of  a 
diiferential  equation. 

The  above  problem  admits  of  a  solution, 
but  we  shall  not  go  into  details  of  the  solu- 
tion. The  interesting  point  to  notice  is  that 
integro-differential  equations  arise  in  a  great 


48  THEORY  OF  PERMUTABLE  FUNCTIONS 

variety  of  problems.  We  have  examined  these 
equations  in  a  number  of  forms  and  have  made 
a  particular  study  of  the  integro-differential 
equations  of  the  second  order  and  of  the  ellip- 
tic or  hyperbolic  types  which  arise  in  connec- 
tion with  certain  problems  of  mathematical 
physics.* 

The  problem  we  were  considering  is  of  a 
different  type.  It  is  of  the  first  order,  and 
since  two  dependent  variables  appear,  it  cor- 
responds to  problems  involving  partial  deriva- 
tives. The  case  of  equation  (11)  may  be 
handled  in  a  similar  manner. 

12.  We  wish  to  demonstrate  certain  inter- 
esting results  which  are  closely  connected 
with  the  problems  we  have  been  discussing. 
Let  us  go  back  to  equation  ( 9 ) .  In  certain 
cases,  this  equation  has  a  finite  number  of  solu- 
tions, while  in  others  the  number  of  solutions 
is  infinite  and  the  solutions  involve  an  arbitrary 
function. 

To  see  this,  we  need  only  to  consider  the 
equation 

*  I.e^'ons  sur  les  fonctions  de  lignes.    Paris:  Gauthier_Villars. 
1913. 


LECTURE  II  49 

and  to  determine  under  what  conditions  x  ~  ^ 
is  the  only  solution  and  under  what  conditions 
solutions  other  than  0  exist. 

To  simplify  matters,  we  shall  assume  that 
the  functions  F  and  ^  are  of  the  first  order 
and  shall  determine  under  what  circimistances 
the  equation  has  a  solution  of  the  first  order. 

Suppose  we  write  oin-  equation  in  the  form 

(B)  flF{:r,^)x{^,f/)d$-{-f'x{.r,^)  ^(^,y)  d^  =  0. 

Then  by  differentiation  with  respect  to  y,  we 
have 

and  when  oj  =  y, 

which  gives  us  a  necessary  condition. 

Moreover,  by  suitable  transformations  of  a 
simple  sort,  we  are  always  led  to  the  case  where 

(12)    F{:x,x)=-^{x,x)  =  l, 

(12')  F,{x,ai)=F,{x,x)  =  ^y{x,x)  =  ^,lx,x)  =  ^ , 

where  the  subscripts  1  and  2  denote  partial 


50  THEORY  OF  PERMUTABLE  FUNCTIONS 

differentiation  with  respect  to  x  and  y  res- 
pectively; that  is 

For  we  can  first  write 

^  =  f(^i) ,      y  =  /(.yi)  , 

X'i^i,//i)=  ±J/'(-n)/'(//i)x(-^.^). 

Then  if  we  take 

_     ±1 
•^'(•"-"^^  ~  F(^) ' 
we  clearly  see  that  the  equation  (B)  becomes 

where 

Hence  we  can  suppose  at  the  outset  that  con- 
ditions (12)  are  satisfied. 

The  above  having  been  established,  equation 


LECTURE  II  51 


(B)  may  be  written 


jy{,^  x{x,  f)  m  ^{^,  u)  ^f  ^^^-=0 


m 
m 

If  we  put 

we  shall  have 

But  we  can  make  use  of  the  arbitrariness  of  a 
and  ^  to  choose 

F\{x,  X)  =  F'lx,  X)  =  ^'i(.i-,  x)  =  ^',{x,  x)  =  0  , 

which  shows  that  we  can  always  assume  that 
condition   (12')   is  satisfied. 
Now  let  us  write 

We  shall  have 


52  THEORY  OF  PERMUTABLE  FUNCTIONS 

whence  we  derive 

where 

/•i(^,^)  =  -^i(.r„//)  +  /f(.^V/)  +  ^i%r,,y)  +  . . . , 

and  therefore  (see  Lecture  II,  §  1) 
fi{x,  x)  =  <f>lx,  x)  =  0  . 
Hence,  integrating  by  parts,  we  have 

where 

and  therefore 

^-g  +  /;  [X(.r,  f )  ^ily)  ->l>{.v,$)  K^.i/)']  <'f =0, 


LECTURE  II  53 

This  integro-differential  equation  may  be  in- 
tegrated. 
If  we  write 

G{x,  y)  =fj  [X(.t-,  I)  i/,(|,  y)  -  xp{:v,  f)  [i{^,  ^)]  d^  , 

we  have 

(16)    i//  (.r,  /j)  =  div)  -  f"  G{^  +  v,v-^)  d^ 
where  d  is  an  arbitrary  function  and 

The  sohition  of  the  equation  (16)  is  obtained 
by  the  method  of  successive  approximations. 

ApjDhcations  of  the  above  will  be  brought 
out  in  the  next  lecture. 


LECTURE  III 


LECTURE  III 

1.  We  shall  begin  with  some  applications 
of  the  work  developed  in  the  last  lecture. 

We  have  solved  the  problem  of  finding  the 
function  x  ('^'^  //)  which  satisfies  the  equation 

on  the  hypothesis  that  F  and  <J>'  are  of  the  first 
order.     Now  suppose  we  put 

Then  the  condition 

<^{x,  X)  +  F{x,  X)  =  0 

is  clearl}^  satisfied,  and  hence  we  shall  be  able 
to  calculate  all  of  the  functions  xi^^V)  which 
satisfy  the  relation 

in  other  words,  all  of  the  functions  which 
have  permutability  of  type  one  with  a  given 
function.  However,  in  the  last  lecture  (§  12) 
this  problem  was  solved  only  in  the  special  case 
where  the  given  function  is  of  the  first  order. 
If  the  function  is  of  higher  order,  the  method 
breaks  down. 


58  THEORY  OF  PERMUTABLE  FUNCTIONS 

We  have  seen  that  the  problem  may  be  re- 
duced to  the  solution  of  an  integro -differential 
equation  of  the  first  order.  If  the  given  func- 
tion is  of  the  second  order,  the  integro-differ- 
ential  equation  which  we  must  solve  is  of  the 
second  order  and  admits  of  a  solution.  An 
arbitrar}^  function  always  enters  in. 

As  we  increase  the  order  of  the  given  func- 
tion, the  problem  becomes  more  and  more  com- 
plicated, hence  we  shall  not  go  into  details  on 
this  question  as  we  should  be  led  too  far  afield. 
In  the  general  case  where  the  functions  are 
analytic  the  question  has  been  answered  by 
M.  Peres.* 

2.  We  wish  to  present  some  of  the  prop- 
erties of  permutable  fimctions.  The  very 
method  which  enables  us  to  calculate  all  of  the 
functions  that  are  permutable  with  a  given 
function  also  leads  us  to  the  result  that  if 
F  and  <I>  are  permutable  and  if  F  is  of  the 
first  order,  then  we  must  have 

^{x,  x)  . 

^    ' — -  =  const. 

F{x,  x) 

We  shall  give  a  rigorous  proof  of  this  fact. 

*  Rendiconti  dci  Lincei.     1913-14. 


LECTURE  III  59 

We  write 

Differentiating  with  respect  to  y,  we  have 

F{x,y)  ^(y,y)+f'F{x,  I)  4>o(|,,y)  d^ 

=  ^(x,//)  FO/,//)  +f'nh  I)  F,{^,//)  d^  , 

and  differentiating  this  last  expression  with 
respect  to  a:, 

F^{x,  y)  ^{y,  y)  +  F{x,  x)  <^^{x,  y) 

-  <I>(:l-,  .T)  F,(x,  y)  +  /J<&i(:*',  I)  F.i^,  y)  d^  . 
Suppose  we  put  <r  =  y.  We  shall  then  have 
Fiiy,  y)  ^{y,  y)  -  F{y,  y)  0,(,y,  y) 

=  ^i{y,  y)  F{y,  y)  -  <^{y,  y)  F.iy,  y) , 
that  is 
lF,{y,y)-^  F.iy,y)]<^(y,y)  = 

F{y,y)[^^y,y)  ^^^iy^y)]  . 
Moreover,  if  we  put 

F{?/,  y)  =  Ay) ,       K!/,  u)=<^  {y) , 

we  have 


60  THEORY  OF  PERMUTABLE  FUNCTIONS 

and  consequently 

whence  the  theorem. 

3.  It  is  a  simple  matter  to  Und  the  expan- 
sion of  any  function  xjj  which  is  permutable 
with  a  function  F  of  the  first  order. 

For  by  the  last  theorem 

\lf{x,  x)  _ 
F{x,x)~^^ 

where  Ci  is  a  constant.    The  expression 

y\f[x,t/)  —  c-i  F{:c,/j) 

will  be  of  higher  order  than  F  and  permut- 
able with  F . 

Now  by  one  of  the  theorems  which  we  proved 
in  the  last  lecture,  w^e  may  write 

where  ^i  will  be  permutable  with  F .     Then 
there  will  be  a  constant  c-z  such  that 

and  consequently  we  shall  have 

>\,  =  c,F  +  c,F~  +  ... 
If  this  process  can  be  carried  on  indefinitelj% 


LECTURE  III  61 

we  shall  have  under  certain  conditions  an  ex- 

* 

pansion  of  i//  in  terms  of  F,  F'^,  .... 

4.  We  shall  give  a  short  survey  of  the 
results  which  can  be  obtained  by  the  intro- 
duction of  a  new  symbol.     If  we  put 

we  can  write 

F=  6-^  ^,        ^  =  ^F-\ 
where  F  ^  and  ^'^  are  merely  symbols  which  do 
not   represent    functions   but   which   may   be 
treated  as  if  thej^  did.     If  the  functions  are 
permutable,  we  can  write 

F  =  ^i-^  =  6-^^, 
and  if 

Fie  =  x, 

hence  the  symbols   <I>'^  and  F^  are  themselves 
permutable. 

Let  us  assume  that  we  have  permutability. 
We  wish  to  determine 


^-1  +  / 


j-i 


62  THEORY  OF  PERMUTABLE  FUNCTIONS 

In  other  words,  let 

We  shall  then  have 

and  owing  to  the  property  of  permutability, 

>i(0i  +  02)  =  {F  +  ^)rjf, 

whence 

01  +  00  =  {F<I>)-'{F^^)^ 

=  {F  +  ^)  {F^)-^^  , 
and  we  may  write 

^_l    _^    j,_i   _    (/^^_<|,)  (i^i)-!   ^ 

that  is,  if/z^  rule  for  the  sum  of  tivo  fractions 
may  he  applied. 

Tlius  we  see  that  we  can  develop  as  it  were 
an  arithmetic  for  the  symbol  F  ^  quite  anal- 
ogous to  the  theory  of  fractions. 

5.  We  have  seen  (§  1)  that  if  ^J*  and  \\t 
are  of  the  first  type  and  if 

^(2-,  .r)  =-  T//(.r, .?;), 
then  a  function  i/;  {a\  if)  may  always  be  found 
such  that 


LECTURE  III  63 

Hence  we  can  write 

(1)      x  =  i-'x^  =  ^xr', 

^  =  x^x~'  ■ 

And  by  solving  the  equation 

we  shall  have  that 

(2)  *  =  y-.xf, 

Therefore  the  two  functions  ^  and  r/» '  can 
always  be  obtained  the  one  from  the  other  by 
a  transformation  through  the  functions  x  or  x'- 

In  particular,  if 

xjj{x;.f)  =  1, 

we  shall  always  be  able  to  find 

(3)  X-iix-'  =  l- 

The  relations  (1)  and  (2)  may  be  obtained 
even  if  ^  and  xjj  are  permutable.  In  this  case, 
X  and  x'  do  not  belong  to  the  group  of  func- 
tions which  are  permutable  with  the  given 
ones.  In  particular,  equation  (3)  may  hold 
even  if  \jj  is  permutable  with  unity. 

6.  We  shall  bring  these  lectures  on  per- 
mutable functions  to  a  close  by  extending  some 


64  THEORY  OF  PERMUTABLE  FUNXTIOXS 

of  the  results  which  were  obtained  in  the  first 
lecture.     (§  11.) 

A  function  which  depends  upon  all  the  val- 
ues of  a  certain  function  /(ct)  between  the 
limits  a  and  h  admits  of  an  expansion 

(4)     A,+j\f[.v,)F,[x,)dx, 

+  r '  r "  ./■(•'•i)  f  (-^'2)  -^2(-^i,  •'^2)  dx,  (h, + . . . , 

provided  certain  conditions  are  satisfied; 
where  F 2(xi,  0C2)  andi^o(cri,  oc-^,  oc^),  etc.,  are 
symmetric  functions.  The  expansion  in  ques- 
tion corresjjonds  to  Taylor's  exj^ansion  (or  to 
a  power  series)  in  ordinary  analysis.* 

With  these  facts  before  us,  let  /'  (.r,  //  |  a) 
be  a  set  of  permutable  functions  of  type  one, 
that  is  of  such  a  sort  that  if  a  be  given  any 
two  values  ai  and  a.^.  the  two  functions  tliereby 
obtained  will  be  permutable  with  one  another. 

As  an  example,  we  give 
./■(•'—// 1  a) 
which  has  the  above  properties. 

*  See:  Legons  sur  les  Equations  integrales  et  intigro-dif- 
ferentielles.  Paris:  Gauthier-Villars.  1913.  Chap.  I,  §  VIII. 
Legons  sur  les  fonctions  de  Ugnes.  Paris:  Gauthier-Villars. 
1913.  Chap.  II.  Lectures  delivered  at  Clark  University, 
Worcester,  Mass.,  1912.     Third  lecture,  §  IV. 


LECTURE  III  65 

Now,  let  US  write 

^     X 

//(•^^li8)/(l,.y»./|=y(.r,^ja,/3). 

The  function  j{oc,  y)  |  a,  ^S)  is  permutable  with 
the  original  ones. 
Again  let  us  write 

and  so  on,  and  let  us  suppose  that  the  series  (4) 
is  convergent  when  \  f  {x)  |  is  less  than  a  cer- 
tain  quantity.     Then  if  we  write  the   series 

Jb     ^,  b 
./'( -f^//  I  -^'l,  •^'2)  ^"(-^'l  v'^-2)  t^-^"l  (^^2  +'", 
a  -^  a 

it  will  converge  no  matter  what  the  absolute 
value  oi  f{oc,y  \  a)  may  be. 
Moreover,  let  us  consider  the  series 

(5)      ^{^)=/{^}  +    f\n.r,)F,{.r,,^)J.v, 

•^  a    ^  a 

If  the  determinant  of  the  linear  integral 


66  THEORY  OF  PERMUTABLE  FUNCTIONS 

equation  which  we  obtain  by  taking  into  con- 
sideration only  the  first  two  terms  does  not 
vanish,  we  can  derive  /(I)  as  a  function  of 
<J>  ( I  )  from  equation  ( 5 ) ,  provided  |  <5»  (  ? )  | 
does  not  exceed  a  certain  vakie.* 
But  let  us  examine  the  series 

« 

Then  if  ^  is  known,  we  can  derive  f{x,y  \  |) 
in  the  form  of  a  series  which  converges  no 
matter  what  the  absolute  value  of  ^  {x,y  \  ^) 
may  be. 

This  is  the  latest  theorem  which  we  have  de- 
rived in  the  field  of  research  we  have  been 
developing. 

*  Leqons  sur  les  Equations  intdgrales  et  integro-diff^rentielles. 
Paris:     Gauthier-Villars.     1913.     Chap.  Ill,  §  XVI. 


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